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: Chapter 7
Migration
Poststack migration
Migration enhances the horizontal (spatial) resolution by:
Moving dipping reflectors to their true subsurface positions and
Collapsing diffractions to their apexes.
Migration is also called seismic imaging.
Migrating a stacked section is called poststack migration, while migrating the pre-stacked data is called prestack migration.
On a stacked section, sources and receivers are coincident.
A reflector is assumed to be directly below its associated source-receiver pair on the stacked section.
This is true for horizontal reflectors; however, dipping reflectors reflect energy at non-vertical direction. Therefore, the above assumption is not true and should be corrected.
In general, a reflection can come from any point on a semicircle whose center is the source-receiver location and radius equal to the traveltime (or depth) to the reflection.
Migration in this case is done by distributing the amplitude at every point on the stacked section over a semicircle. Constructive interference of these semicircles produces the stacked migrated section.
Alternatively, we can think of every point in the subsurface as a point scatterer that produces a diffraction hyperbola on the stacked section whose apex is at the position of the scatterer. Constructive interference of these hyperbolae produces the stacked unmigrated section.
Migration in this case is done by constructing a hyperbola at every point on the stacked section and summing the amplitudes lying over each hyperbola and assigning the sum at its apex. The result of this process is the stacked migrated section.
Prestack migration
Prestack migration takes into account the location of the source and receiver for each trace when determining the reflector position.
Before stack, a reflection can come from any point on an ellipse whose foci are the source and receiver.
Prestack migration is done by spreading the amplitude at every point in the prestack gathers over an ellipse. Constructive interference of these ellipses will produce the prestack migrated section.
Time migration versus depth migration
Diffractions are hyperbolic only if there are no lateral heterogeneities because they can distort the shape of diffractions.
Time migration assumes hyperbolic diffractions and collapses them to their apexes.
Depth migration assumes a known velocity model and estimates the correct shape of diffractions by ray tracing or wavefront modeling.
Time migration produces a migrated time section while depth migration produces a migrated depth section.
Evidently, using time migration followed by time-to-depth conversion does not produce a depth-migrated section.
Time migration is valid only when lateral velocity variations are mild to moderate. When this assumption fails, we use depth migration.
Migrated sections are commonly displayed in time rather than depth for the following reasons:
To avoid errors introduced by inaccurate time-to-depth conversion.
To facilitate the comparison of migrated sections with unmigrated sections, which are usually displayed in time.
2-D migration versus 3-D migration
In 2-D migration, we migrate the data once along the profile. This might generate misties on intersecting profiles.
In addition, 2-D migration is prone to sideswipe effects. Sideswipes are reflections from out of the plane of the profile.
In 3-D migration, we first migrate the data in the inline direction then take that migrated data and migrate it again in the crossline direction. This is the two-pass 3-D migration.
One-pass 3-D migration can also be done using a downward continuation approach.
Therefore, considering 2-D versus 3-D, prestack versus poststack, and time versus depth, we can have the following types of migrations (ordered from fastest but least accurate to slowest but best accurate):
2-D poststack time migration.
2-D poststack depth migration.
2-D prestack time migration.
2-D prestack depth migration.
3-D poststack time migration.
3-D poststack depth migration.
3-D prestack time migration.
3-D prestack depth migration.
Geometrical aspects of migration
Before migration, synclines look like bowties because of the interference among diffraction hyperbolae. These bowties are untied into synclines after migration.
Graphically, a linear reflector can be migrated by the following procedure:
Select two points on the reflector.
Draw two semicircles whose centers are the source-receiver pair of each point and radii are equal to the TWTT to these points.
The common tangent to these semicircles is the migrated reflector.
When a dipping reflector is migrated, it is moved updip, steepened, and shortened.
The amount of horizontal displacement (dx), vertical displacement (dt) and dip (Dt/Dx) introduced by the migration are given by the following relations:
EMBED Equation.3 , (8.1)
EMBED Equation.3 , (8.2)
EMBED Equation.3 , (8.3)
where v: medium velocity, t: traveltime on unmigrated section, t : traveltime on migrated section, and Dt/Dx is the dip on unmigrated section.
dx, dt, and Dt/Dx increase with time, velocity, and dip of reflector on the unmigrated section.
Another important relation is the migrator s equation given by:
sinb = tana, (8.4)
where a is dip angle on unmigrated section and b is dip angle on migrated section.
From equation (8.4), we can see that b ( a and that the maximum dip on the unmigrated section is 45(, which is the dip of an asymptote of a diffraction hyperbola.
Migration of noise usually produces migration smiles due to the non-interference of the semicircles.
Migration techniques (algorithms)
There are three main migration algorithms:
Kirchhoff migration: for every point on the unmigrated section:
Construct a diffraction hyperbola.
Sum the amplitudes of samples lying on this hyperbola after scaling.
Assign the summed amplitude at the apex of the hyperbola.
F-K (Stolt) migration:
This method makes use of the 2-D Fourier transform to convert the input unmigrated section (xu,tu) into the 2-D Fourier (fu,ku) domain where it is migrated with a simple algorithm to produce (fm,km).
The inverse transform of (fm,km) provides the migrated section (xm,tm).
F denotes frequency (the Fourier transform of time) while K denotes wavenumber (the Fourier transform of space or distance).
Finite-difference (downward continuation) migration:
This method works on a conceptual volume of information (x,z,t) rather than two time (x,t) and depth (x,z) planes.
The method can be summarized in the following steps:
Input the top surface seismic section (x,z=0,t).
Compute the entire volume (x,z,t).
Extract the structure (x,z,t=0).
Kirchhoff migration
This method employs Huygens principle, which states that every point on the wavefront can be regarded as a secondary source that generates seismic waves in the forward direction.
Huygens secondary sources are point apertures, which produce waves that depend on propagation angle; unlike point sources, which are isotropic.
A Huygens secondary source generates a semicircular wavefront in the (x,z) plane and a diffraction hyperbola in the (x,t) plane.
The purpose of Kirchhoff migration is to sum up the energy produced by every Huygens secondary source and map it into its point of generation.
Therefore, there are two schemes for Kirchhoff migration:
Superposition of semicircular wavefronts (Hagedoorn or distribution migration).
Superposition of diffraction hyperbolas (summation migration).
The semicircle superposition approach was used earlier and abandoned because of:
its unsuitability for computer implementation and
its invalidity for laterally variable velocity structures.
Kirchhoff migration is based on the far-field solution of the 3-D scalar wave equation:
EMBED Equation.3 , (8.5)
where Pout is the migrated amplitude, Pin is the unmigrated amplitude, ( is the propagation angle from the vertical, v is the RMS velocity at the point to be migrated, and t is the vertical TWTT at the point to be migrated.
cosq is called the obliquity factor, which decreases the amplitude along the hyperbola as we get away from the apex.
1/v2t is the spherical divergence factor due to geometrical spreading.
(Pin/(t is called the wavelet shaping factor and it is equivalent to applying a 90( phase-shift to the wavelet and multiplying the amplitude spectrum by its corresponding frequency.
The integration is carried over the width (ymin,ymax) of the diffraction hyperbola whose apex is located at the point to be migrated (i.e., y = 0 corresponds to the location of the apex of the hyperbola).
Therefore, the Kirchhoff migration algorithm amounts to applying the following steps (for each point on the unmigrated zero-offset time section):
Computing the hyperbolic traveltime as:
t2(y) = t02 + 4y2/v2, (8.6)
where t(y) is TWTT at a distance y from the point to be migrated (hyperbola
apex), t0 is the vertical TWTT from the surface to the point to be migrated, and
v is the RMS velocity at the point to be migrated, respectively.
Applying the obliquity factor.
Applying the spherical divergence factor if it has not been applied already.
Applying the wavelet shaping factor.
Summing the resultant amplitudes along the hyperbola for all possible offsets.
Assigning the sum at the apex of the hyperbola (i.e., point to be migrated).
Kirchhoff migration in practice
The main important parameters when applying Kirchhoff migration in practice are:
Aperture width.
Maximum dip to migrate.
Velocity errors.
Aperture width:
Migration aperture (apron) is the spatial extent over which migration is carried out.
In terms of hyperbolic summation, the aperture is the maximum width of the hyperbola over which the summation is carried out.
In terms of dip, the aperture is given by the horizontal displacement dx as defined by equation (8.1).
Excessively small aperture widths produces the following undesirable effects:
It suppresses steeply dipping events because migration is only using the flat part of the hyperbola that corresponds to small dip.
It organizes random noise as horizontal events, especially in the deeper part of the section.
Excessively large aperture widths produces the following undesirable effects:
It needs more, sometimes unnecessary, computer time.
It degrades the migration quality in poor S/N ratio conditions because more noisy data from the deeper part of the section will be included.
It is recommended that the aperture is kept constant in migrating all lines from a particular survey so that an overall uniformity in amplitude characteristics on the migrated sections is maintained.
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h7@7^7`gd;In practice, a regional velocity function and the maximum dip in the survey area are used to compute the optimal aperture width that can be used over the entire set of data using equation (8.1).
For any given time, the optimal value of the aperture width is defined by twice the maximum horizontal displacement in migration for the steepest dip of interest in the input section.
Maximum dip to migrate:
The maximum dip to migrate is related to the aperture width by equation (8.1).
Smaller maximum dip limit means smaller aperture.
Large maximum dip limit means more computer time.
The maximum dip can be used to filter steeply dipping coherent noise.
Velocity errors:
Velocity affects the horizontal displacement, as seen from equation (8.1).
Using a lower velocity produces undermigration due to the incomplete collapse of diffraction hyperbolae.
Using a higher velocity produces overmigration due to the generation of reversed diffractions.
Steeper dips are more sensitive to velocity errors, as seen from equation (8.3).
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